engineering reliability - failure models

ENGINEERINGRELIABILITY

INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

ENGINEERING RELIABILITYFAILURE MODELS

Harry G. Kwatny

Department of Mechanical Engineering & MechanicsDrexel University


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

OUTLINE

INTRODUCTION

FAILURE RATETime to FailureReliability FunctionFailure RateMTTF & MRL

CONSTANT RATE MODELSThe Exponential DistributionRepeated Demand

VARIABLE RATE MODELS

SUMMARY


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

INTRODUCTION

We will be concerned with the following measures of aproduct or system that is not repaired upon failure.

I The reliability (or survivor) function, R (t).I The failure rate, z (t) or λ (t).I The mean time to failure, MTTF.I The mean residual life MRL.I Constant rate models.


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

THE MEANING OF TIME

The failure time T of a product or systems is a randomvariable. Time can take on different meanings depending onthe context, e.g.,

I Calender time.I Operational time.I Distance driven by a vehicle.I Number of cycles for a periodically operated system.I Number of times a switch is operated.


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

PROBABILITY CHARACTERIZATION OF FAILURE

TIME

Associated with the time to failure T is the probability function

F (t) = P (T ≤ t)

which is the probability that the system fails within the timeinterval (0, t]. If T is a continuous random variable, the probabilityfunction is related to its probability density function f (t) by

F (t) =∫ t

0f (τ)dτ

0.5 1 1.5 2 2.5 3t

0.20.40.60.81

fHtL,FHtL


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

LOGNORMAL DISTRIBUTION

I The lognormal distribution has been found useful in the failure analysis ofitems subjected to repeated loadings

I while the normal distribution is ideal for characterizing the influence of the

sum of a large number of independent events, the lognormal is appropriate

for characterizing the product of a large number of independent events

f (t) =1√

2πσte−

12 ( ln t−µ

σ )2

, t > 0

F (t) =12

(1 + Erf

(ln t − µ√

2σ

)), t > 0

2 4 6 8 10t

0.2

0.4

0.6

0.8

1fHtL

0.9σ =0.5σ =

0.3σ =

0.15σ =


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

RELIABILITY FUNCTION

The reliability function is:

R (t) = P (T > t) = 1− F (t) =∫ ∞

tf (τ) dτ

I R (t) is the probability that the item will not fail in theinterval (0, t].

I R (t) is the probability that it will survive at least untiltime t – it is sometimes called the survival function.

0.5 1 1.5 2 2.5 3t

0.20.40.60.81RHtL


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

FAILURE RATE

Consider the conditional probability:

P (t < T ≤ t + ∆t |T > t ) =P (t < T ≤ t + ∆t)

R (t)

=F (t + ∆t)− F (t)

R (t)The failure rate (or, hazard function) is defined as:

λ (t) = lim∆t→0

P (t < T ≤ t + ∆t |T > t )

∆t=

f (t)R (t)

λ(t) dt is the probability that the system will fail during the period(t, t + dt], given that it has survived until time t.

0.25 0.5 0.75 1 1.25 1.5 1.75 2t

0.250.50.75

11.251.51.75

2zHtL

Burn in Wear outConstant rate


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

DISTRIBUTIONS FROM FAILURE RATE

Suppose the failure rate λ (t) is known. Then it is possible toobtain f (t), F (t), and R (t)

f (t) =dF (t)

dt= −dR (t)

dt⇒ λ (t) = −dR/dt

R

dRR

= −λ (t) dt

⇓

R (t) = exp[−∫ t

0 λ (τ) dτ]

f (t) = λ (t) exp[−∫ t

0 λ (τ) dτ]

F (t) = 1− exp[−∫ t

0 λ (τ) dτ]


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

EXAMPLE – TV SETS

EXAMPLE (TV SET FAILURE DATA)

Day failures failure rate1 18 0.0182 12 0.0123 10 0.0104 7 0.0075 6 0.0066 5 0.0057 4 0.0048 3 0.0039 0 0

10 1 0.001


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

EXAMPLE – TV SETS, CONT’D

00.0020.0040.0060.0080.01

0.0120.0140.0160.0180.02

1 2 3 4 5 6 7 8 9 10

DaysF

ailu

re R

ate

0.5 1 1.5 2t

0.05

0.1

0.15

0.2λHtL−,fHtL−−

5 10 15 20t

0.005

0.01

0.015

0.02λHtL−,fHtL−−

λ (t) = 0.02 t−0.56 f (t) = 0.02 t−0.56e−0.04545t0.44


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

WEIBULL DISTRIBUTION

Perhaps the most frequently used distribution to model timeto failure probabilities is the Weibull distribution,Weibull(α, β), The probability function is

F (t) ={

1− e−(βt)α t ≥ 00 t < 0

and the corresponding density function is

f (t) =ddt

F (t) ={αβαtα−1e−(βt)α t ≥ 0

0 t < 0

0.5 1 1.5 2 2.5 3t

0.2

0.4

0.6

0.8

1

1.2

1.4

fHtL

0.5α =

1.0α =

2.0α =

3.0α =


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

WEIBULL DISTRIBUTION – (2)

The Reliability function is:

R (t) = 1− F (t) = e−(βt)α , t > 0

and the failure rate function is

λ (t) =f (t)R (t)

= αβαtα−1, t > 0

0.2 0.4 0.6 0.8 1t

0.25

0.5

0.75

11.25

1.51.75

2λHtL

0.5α =

1.0α =

2.0α =

3.0α =


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

MEAN TIME TO FAILURE

The mean time to failure is

MTTF = E {T} =∫ ∞

0t f (t) dt

note thatf (t) =

ddt

F (t) = − ddt

R (t)

from which

MTTF = −∫ ∞

0t

dR (t)dt

dt = − tR (t)|∞0 +∫ ∞

0R (t) dt

MTTF =∫ ∞

0R (t) dt


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

EXAMPLE

EXAMPLEConsider a system with reliability function

R (t) =1

(0.2t + 1)2 , for t > 0 (t in months)

I probability density f (t) = − ddt R (t) = 0.4

(0.2t+1)3

I failure rate λ (t) = f (t)R(t) = 0.4

(0.2t+1)

I mean time to failure MTTF =∫∞

0 R (t) dt = 5 months


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

MEAN RESIDUAL LIFE

An item begins operation at time 0 and is still operating attime t. We wish to compute the probability that it will survivean additional interval of length τ . This is the conditionalreliability function at age t.

R (τ |t ) = P (T > t + τ |T > t ) =P (T > t + τ)

P (T > t)=

R (t + τ)R (t)

The mean residual (or, remaining) life at age t is

MRL (t) =∫ ∞

0R (τ |t ) dτ =

1R (t)

∫ ∞t

R (τ)dτ


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

EXAMPLE

Consider an item with failure rate λ (t) = t/(t + 1) Nowcompute

R (t) = exp(−∫ t

0

τ

τ + 1dτ)

= (t + 1) e−t

MTTF =∫ ∞

0(t + 1) e−t dt = 2

The conditional reliability function is

R (τ |t ) = P (T > τ + t |T > τ ) =(t + τ + 1) e−(t+τ)

(t + 1) e−t =t + τ + 1

t + 1

So,

MRL =∫ ∞

0R (τ |t ) dτ = 1 +

1t + 1


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

EXAMPLE: WEIBULL DISTRIBUTION

The mean time to failure is

MTTF =∫ ∞

0R (t) dt =

1β

Γ(

1α

+ 1)

The remaining life is

MRL =1

R (t)

∫ ∞t

R (t) dt =(α

β

)e( t

β )α

Γ(

1α,

(tβ

)α)The median life is

R (tm) = 0.50⇒ tm =1β

(ln 2)1/α

The variance of T is

var (T) =1β2

(Γ(

2α

+ 1)

+ Γ2(

1α

+ 1))


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

THE EXPONENTIAL DISTRIBUTION

Suppose λ (t) = λ0, a constant. Then,

R (t) = e−λ0t, F (t) = 1− e−λ0t, f (t) = λ0e−λ0t

This is the exponential distribution. We can easily compute

µ = MTTF =1λ0

σ =1λ0

R (µ) = 0.368, F (µ) = 0.632


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

EXAMPLE

Suppose λ = .02 hr−1,

I What is the probability of a failure in the first 10 hours ofservice?

P (T 6 10) = F (10) = 1− e0.02×10 = 0.181

I Suppose the unit operates satisfactorily for the first 100hours. What is the probability of failure in the next 10 hours?

I Let X = event that the unit operates for 100 hours⇒ P (X) = R (100) = .135

I Let Y = event that the unit fails within 110 hours⇒ P (Y) = F (110) = .1108

I We want to compute P (Y|X). By definition

P (Y|X) =P (Y ∩ X)

P (X)=

P (100 6 t 6 110)P (X)

=F (110)− F (100)

R (100)= 0.181


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

UNIT UNDER REPEATED DEMAND

Suppose a unit is subjected to repeated demand (e.g.,engine startup) over a large time interval (0, t]:

I p denotes the probability of failure to respond to a demand,I the response to each demand is an independent event,I n denotes the number of demands in time t,I m = n/t denotes the average number of demands per unit

time,

The probability of k failures in n demands is given by theBinomial distribution b (k; n, p) = Ck

npk(1− p)n−k.Let N denote the number of trials until the first failure. Thus,N = n means that the first n− 1 trials are successful, andfailure occurs at trial n. The distribution of N is the geometricdistribution

P (N = n) = (1− p)n−1p


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

UNIT UNDER REPEATED DEMAND, CONT’D

The reliability function, R(n), is the probability that the firstfailure occurs for some trial N > n. Consequently,

R (n) = P (N > n) = b (0; n, p) = (1− p)n

Suppose the single trial failure probability p is small and thelength of the observed sequence N large, in fact Np = λ,λ = constant. Then,

limp→0

(1− p)λp = e−λ ⇒ R (n) = e−np = e−mpt = e−λ0t

where λ0 = mp is the equivalent failure rate.


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

MULTIPLE PERFORMANCE LEVELS

A unit operates at different performance levels during cyclicoperation. In each operating phase the unit fails at constantrate.

EXAMPLEI a motor cycles through 3 phases: start, run, standby,I N, number of starts per service cycle,I T denotes the number of service cycles to failure,I c, time fraction motor runs during a cycle,I 1− c, time fraction motor is in standby,I p, probability of failure to start,I λr, failure rate in run state,

I λs, failure rate in standby state.

λc , λd + cλr + (1− c)λs, where λd , Np

R (t) = e−λct


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

COMMON VARIABLE RATE MODELS

I Time to failure is typically described by normal,lognormal or Weibull probability distributions.

I The corresponding failure rates can be computed from

λ (t) = − 1R (t)

dR (t)dt

Normal : λ (t) =φ (z)

σ (1− Φ (z)); z =

t − µσ

Lognormal : λ (t) =φ (z)σtΦ (z)

; z =ln (t)− µ

σ

Weibull : λ (t) = (αβ) (βt)α−1


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

THE WEAR-IN FAILURE MODE & THE PROOF

TEST

I When initial failure rates are high, testing can improve thereliability of deployed product.

I An initial, short period, tp, of testing of the entire batch weedsout faulty product.

I suppose λ (t) = at−b, a, b > 0, the we can computeR (t − tp|tp)

R (τ |tp) =R (tp + τ)

R (tp)=

e−∫ tp+τ

0 λ(ξ)dξ

e−∫ tp

0 λ(ξ)dξ= e−

∫ tp+τtp

λ(ξ)dξ

R(t)

t

1

tp

R(tp)

Early failures

tp+

R(tp+ )

R( tp|tp)


INTRODUCTION

FAILURE RATE

TIME TO FAILURE

RELIABILITY

FUNCTION

FAILURE RATE

MTTF & MRL

CONSTANTRATE MODELS

THE EXPONENTIAL

DISTRIBUTION

REPEATED DEMAND

VARIABLERATE MODELS

SUMMARY

SUMMARY

I Time to failureI Failure rateI Computation of probability functions from failure rateI definitions of mean time to failure (MTTF) and

remaining life (MRL)I Introduced lognormal, exponential and Weibull

distributionsI Examples of constant failure rate problemsI Proof Testing

engineering reliability - failure models

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